On L-equivalence for K3 surfaces and hyperk\"ahler manifolds

This paper explores the relationship between L-equivalence and D-equivalence for K3 surfaces and hyperk\"ahler manifolds. Building on Efimov's approach using Hodge theory, we prove that very general L-equivalent K3 surfaces are D-equivalent, leveraging the Derived Torelli Theorem for K3 surfaces. Our main technical contribution is that two distinct lattice structures on an integral, irreducible Hodge structure are related by a rational endomorphism of the Hodge structure. We partially extend our results to hyperk\"ahler fourfolds and moduli spaces of sheaves on K3 surfaces.